Milnor’s Exotic 7-Spheres

Jhan-Cyuan Syu (許展銓)

June, 2017

0 Introduction

This article is devoted to the explicit construction of Milnor’s exotic 7-spheres,

together with some prerequisites needed in the construction, such as characteristic

classes, cobordisms and signatures.

There will be a nature question rising immediately as soon as we define a smooth

structure on a manifold: can a topological manifold admit two different smooth

structure? That is, is it true that if two smooth manifolds are homeomorphic, then

they are automatically diffeomorphic? The answer is that it depends!!

If the manifold is of dimension 1 or 2, then the answers are sure, proved by

Tibor Radó. The case of dimension 3 was also proved to be true by Edwin E.

Moise. However, the first counterexample given by John Milnor in 1956 states

that there are at least two different smooth structures on 7-sphere. Furthermore,

John Milnor and Michel Kervaire proved that there are 28 oriented differentiable

structures on 7-sphere (15 if without consideration of orientation) in 1963. As time

passed by, Michael Freedman gave a non-standard differentiable structure on R4,

known as exotic R4, in 1982. By the way, there are no different smooth structures

on Rn for all n = 4. Even though exotic R4 has been constructed over 30 years ago,

the existence of exotic 4-sphere is still an open problem now.

1

1 Fiber Bundles and Characteristic Classes

Given a real vector bundle ξ : Eπ−→ M of rank 2n, one can give each fiber a complex

structure in the following sense: J : E → E is a continuous map which maps each

fiber R-linearly to itself and satisfies J J(v) = −v for all v ∈ E. With this complex

structure, the real vector bundle ξ can then be viewed as a complex vector bundle.

Conversely, given a complex vector bundle ξ, we can also view it as a real vector

bundle, denoted by ξR, by ignoring the complex structure on each fiber and think

of each fiber as a real vector space.

Let ξ : Eπ−→ M be a complex vector bundle over a manifold M . We define

ξ : Eπ−→ M the complex conjugate vector bundle of ξ by the following way:

(i) ξR = ξR.

(ii) The identity map i : E → E is conjugate linear, i.e., i(cv) = ci(v) for each c ∈ C

and v ∈ E.

Now we are ready for the definition of our first characteristic class: Chern

classes.

Definition. (Chern Classes)

Let ξ : Eπ−→ M be a complex vector bundle of rank n (complex dimension) over a

manifold M . The total Chern class of this bundle is

c(ξ) =∑i≥0

ci(ξ) ∈ H2n(M,Z)

satisfying

(i) c0(ξ) = 1, ci(ξ) ∈ H2i(M,Z) for all i and ci(E) = 0 for i > n.

(ii) Naturality: for any smooth map f : M → M ′ from M to another manifold M ′,

c(f ∗ξ) = f ∗c(ξ).

(iii) Whitney sum formula: c(ξ ⊕ η) = c(ξ) c(η) for any complex vector bundle η

over M .

(iv) c(γ) = 1+ g, where γ is the universal line bundle of CP∞ and g ∈ H2(CP∞) is

the generator of the cohomology.

With definition of Chern classes, we define the Euler class e(ξR) = cn(ξ) ∈

H2n(M,Z), where ξ is a n-dimensional complex vector bundle over M , and the total

2

Pontryagin class p(η) =∑

i∈N0pi(η) with pi(η) = (−1)ic2i(η ⊗ C) ∈ H4i(M,Z),

where η is a real vector bundle over M .

There are two facts that indicate us how to compute Pontryagin classes from

Chern classes.

Fact. If ξ is a real vector bundle, then ξ ⊗ C ∼=C ξ ⊗ C; if ξ is a complex vector

bundle, then ξR ⊗ C ∼=C ξ ⊕ ξ.

Fact. ci(ξ) = (−1)ici(ξ).

Now we recall some basic definitions and properties of homotopy groups, which

will frequently appear throughout the whole article.

Definition. (Homotopy Groups)

Define πn(X, x0) = f : Sn → X | f is continuous and f(t0) = x0/ ∼, where t0 is

some fixed point of Sn and the equivalence relation ∼ is the homotopic equivalence.

The composition law is [f ] + [g] := [f + g].

For two continuous maps f, g : Sn → X with f(t0) = g(t0) = x0, we de-

fine (f + g) : Sn → X to be the composition of two maps Ψ ρ, where ρ : Sn →

Sn ∨Sn ∼= Sn ×t0∪t0×Sn and Ψ : Sn∨Sn → X. The space Sn∨Sn is obtained

by choosing an equator of Sn passing through t0 and then identifying the equator as

one point, so the resulting space has the isomorphism Sn∨Sn ∼= Sn×t0∪t0×Sn

and ρ : Sn → Sn ∨Sn is defined by sending the upper sphere to Sn×t0, the lower

sphere to t0 × Sn and the equator to t0 × t0. The map Ψ : Sn∨Sn → X is

given by Ψ(· × t0) = f(·) and Ψ(t0 × ·) = g(·).

3

We will next introduce two important examples of vector bundles, which both

play crucial roles in the following story.

Example. (Hopf Quaternion Bundles)

Let H := a + bi + cj + dk | a, b, c, d ∈ R the quaternions. Let q0, · · · , qn

be the standard basis of Hn+1. We now consider S4n+3. Indeed, S4n+3 can be

embedded into Hn+1 ∼= R4n+4, i.e., S4n+3 → R4n+4 ∼= Hn+1, so we can represent

S4n+3 in quaternion coordinate:

S4n+3 := (q0, · · · , qn) ∈ Hn+1 | |(q0, · · · , qn)|2 =n∑

α=0

|qα|2 = 1.

Then there is a natural action of SU(2) ∼= S3 → H on S4n+3 given by

q · (q0, · · · , qn) := (qq0, · · · , qqn) ∈ S4n+3 → Hn+1,

the left multiplication in quaternions by q ∈ H with |q| = 1. Hence we obtain a

principal SU(2)-bundle with fiber SU(2) ∼= S3:

SU(2) S4n+3

S4n+3 / SU(2)

action

π

where S4n+3 / SU(2) ∼= S4n+3 /S3∼= HPn.

Let γn be the universal line bundle of HPn and define

un := c2(γnC) = e(γnR) ∈ H4(HPn,Z).

By CW-decomposition of HPn, one can conclude:

(i) H4i(HPn,Z) ∼= uin Z.

(ii) H i(HPn,Z) = 0 if 4 - i.

So c(γnC) = 1 + c1(γnC) + c2(γnC) = 1 + un.

For p(γnR), notice that c0(γnC) = 1, c2(γnC) = u and ci(γnC) = 0 for i = 0, 2

and compute

p(γnR) =∑i∈N0

pi(γnR) =∑i∈N0

(−1)ic2i(γnR ⊗ C) =∑i∈N0

(−1)ic2i(γnC ⊕ γnC)

= c0(γnC)c0(γnC)− c0(γnC)c2(γnC)− c2(γnC)c0(γnC) + c2(γnC)c2(γnC)

= 1− un−un +u2n = 1− 2un+u2

n .

4

Example. (SO(4)-Vector Bundles over S4 of Rank 4)

We first recall a useful lemma:

Lemma. The G-bundles over Sk are determined by πk−1(G).

There is an intuitive explanation of this lemma: View Sk as the union of the

upper sphere and the lower sphere. Each half sphere is contractible and thus their

vector bundles are both trivial. Therefore, to obtain a vector bundle over Sk, we

should glue the two trivial bundles along the fiber of the equator, and each distinct

approach to gluing up two bundles gives distinct vector bundles over Sk.

To study SO(4)-vector bundles over S4, we should look at π3(SO(4)). Since

π3(SO(4)) ∼= π3(SO(3))⊕ π3(S3) ∼= Z⊕Z,

we know that SO(4)-vector bundles over S4 are determined by integer pairs (h, j) ∈

Z2.

Define fhj : S3 → SO(4) by fhj(v)w := vhwvj, where v, w ∈ H and v ∈ S3 → H.

Let ξhj be the vector bundle defined by fhj with fiber R4 and Shj the sphere bundle

of ξhj.

Define σ : S3 → SO(4) and σ′ : S3 → SO(4) by σ(v)w := vw and σ′(vw) := wv,

the left and right multiplication in H. Since HP1 ∼= S4, so this is a special case

of Hopf quaternion bundles with n = 1. Let γ := γ1 be the left Hopf bundle

corresponding to σ discussing in last example and γ′ the vector bundle corresponding

to σ′. Then σ and σ′ generate π3(SO(4)).

Proposition 1. e(ξhj) = (h+ j)u and p1(ξhj) = 2(h− j)u, where u := u1.

Proof. As noted above, this is a special case of Hopf quaternion bundles with n = 1.

In this case, e(γR) = u and

p1(γR) = −c2(γC ⊗ C) = −c2(γC ⊕ γC) = −[c0(γC)c2(γC) + c2(γC)c0(γC)] = −2u .

Similarly, e(γR) = u and p1(γR) = 2u. Then

e(ξhj) = (h+ j)u and p1(ξhj) = 2(h− j)u

follow from definition.

5

Example. (Oriented Canonical Vector Bundles of Grassmannians)

Let Gk(Rk+p) be the Grassmannian consisting of all k-dimensional subspaces

of Rk+p. Further, define Gk(Rk+p) be the oriented Grassmannian, i.e., its elements

are those oriented k-dimensional subspaces of Rk+p. Given Gk(Rk+p), there is a

canonical vector bundle

γkp ≡ γk(Rk+p) := (X, v) | X ∈ Gk(Rk+p) and v ∈ X.

Similarly, define oriented canonical vector bundle of Gk(Rk+p) by

γkp ≡ γk(Rk+p) := (X, v) | X ∈ Gk(Rk+p) and v ∈ X.

We now introduce two notations about partition numbers:

P(n) := (i1, · · · , ir) ∈ Nr | r ∈ N, i1 ≤ · · · ≤ ir, i1 + · · ·+ ir = n

p(n) := |P(n)|, the cardinality of P(n).

Theorem 1. Let R be a ring with 2 being invertible. ξm denotes the universal bundle

over G2n−1(R∞) of rank m. Then

H∗(G2n+1(R∞);R) = R[p1(ξ2n+1), · · · , pn(ξ2n+1)]

H∗(G2n(R∞);R) = R[p1(ξ2n), · · · , pn−1(ξ

2n), e(ξ2n)].

6

2 Thom Spaces and Cobordisms

Let M be an oriented manifold. We denote by −M the same manifold with opposite

orientation. For another oriented manifold M ′, M +M ′ denotes the disjoint union

of M and M ′.

Definition. (Cobordism Groups)

We say that two smooth oriented compact manifolds M1,M2 both of dimension n

are (oriented) codordant if M1+(−M2) = ∂W for some (n+1)-dimensional smooth

oriented compact manifold-with-boundary W . In this case, we say that M,M ′ are

in the same cobordiam class, denoted by [M ] = [M ′].

The n-th (oriented) cobordism (group) Ωn is the set of all (oriented) cobordism

classes of dimension n.

The above definition indeed make sense because the relation of cobordism

classes is clearly an equivalence relation. Another fact is that Ωn is an abelian

group. Our attention will turn to the direct sum of all cobordisms: Ω :=⊕

n∈N0Ωn.

Recall that a graded ring is a direct sum of abelian groups Gα such that

Gα × Gβ ⊂ Gα+β. We check that Ω is definitely a graded ring. The only thing we

have to verify is that the map Ωi × Ωj → Ωi+j given by ([M ], [N ]) 7→ [M × N ] is

well-defined. Let [M ] = [M ′] ∈ Ωi and [N ] = [N ′] ∈ Ωj. Write M −M ′ = ∂W and

N −N ′ = ∂V . Then

M ×N −M ′ ×N ′ = (M ′ + ∂W )×N −M ′ × (N − ∂V )

= ∂W ×N +M ′ × ∂V

= ∂(W ×N −M ′ × V )

Hence [M ]× [N ] = [M ′]× [N ′].

Definition. (Transversality)

Let M,N be two smooth manifolds and N ′ a submanifold of N . f : M → N is a

smooth map. For a subset A of M we say f is transverse to N ′ over A if for each

p ∈ A ∩ f−1(N ′) the following condition holds:

df(TpM) + Tf(p)N′ = Tf(p)N.

7

In this case, we denote it by f tA N ′. In particular, we write f t N ′ if A = M .

If f t y, then we called y ∈ N is a regular value of f .

In other words, f tA N ′ if and only if the composition of maps

TxMdfx−→ Tf(x)N → Tf(x)N/Tf(x)N

′

is surjective for all x ∈ A ∩ f−1(N ′). Specifically, if N ′ = y is a set containing

only one point, then dfx is surjective for all x ∈ A ∩ f−1(y).

Fact. Let f : M → N be a smooth map and y ∈ N a point. Assume dimM = m

and dimN = n. If f t y, then f−1(y) is a smooth manifold of dimension m− n.

Proof. Let x ∈ f−1(y). Since the map dfx : TxM → TyN is surjective, we then

have N := ker dfx is a smooth manifold of dimension of m−n. Embed M into Rk for

some k. Choose a linear map L : Rk → Rm−n to be non-singular on N ⊂ TxM → Rk.

Define

F : M → N × Rm−n

x 7→ (f(x), L(x)).

Its differential is given by dFx(v) = (dfx(v), L(v)), so dFx is surjective. Hence

F maps some neighborhood U of x diffeomorphically to some neighborhood V of

(y, L(x)). Therefore, F maps f−1(y) ∩ U diffeomorphically to (y × Rm−n) ∩ V .

That is, f−1(y) is a smooth manifold of dimension m− n.

Fact. (Brown’s Theorem)

Let f : M → N be a smooth map. Then the set of regular values of f is dense in

N , i.e., the set y ∈ N | f t y is dense in N .

Proof. This is a corollary of Sard’s theorem. We first recall the statement of ssard’s

theorem.

Sard’s Theorem. Let f : M → N be a smooth map between manifolds M,N .

Then the set

C := x ∈ M | dfx is not surjective.

8

has (Lebesgue) measure 0 in N.

Then Brown’s theorem follows from N − f(C) = y ∈ N | f t y and

N − f(C) is dense in N .

The task for us is to approximate a map transverse to 0 over a closed subset

by a map transverse to 0 over a larger closed subset.

Lemma 1. Suppose M is an open subset of Rm. Let X ⊂ M be a closed subset

in M and K a compact subset of M . Suppose f : M → Rn is a smooth map and

f tX 0. Fix a compact K ′ ⊂ M with K ⊂ (K ′ − ∂K ′). Given ε > 0 then there

exists a smooth map g : M → Rn satisfying:

(i) g tX∪K 0.

(ii) f |cK′= g |cK′, where cK ′ := M −K ′.

(iii) |f(x)− g(x)| < ε for all x ∈ M .

Proof. Let λ : M → [0, 1] be a smooth cut-off function such that λ |K≡ 1 and

λ |cK′≡ 0. According to the fact mentioned above, we can take y ∈ Rn with |y| < ε

such that f t y. Define g(x) = f(x)− λ(x)y and check that

(a) f |cK′= g |cK′ .

(b) |f(x)− g(x)| < ε for all x ∈ M .

(c) g tK 0.

(a) and (b) are obvious. For (c), if x0 ∈ g−1(0) ∩ K, then 0 = g(x0) = f(x0) −

λ(x0)y = f(x0)− y =⇒ x0 ∈ f−1(y). By our assumption, f t y, i.e.,

df(Tx0M) = Ty Rn = T0Rn .

In addition,

dg(Tx0M) = d(f + λy)(Tx0M) = df(Tx0M) = T0Rn .

Hence we verify that (c) holds.

Claim: If y is chosen to be sufficiently close to 0, then g tK′∩X 0.

With this claim, together with (b) and (c), we then prove g tX∪K 0.

Now we prove the claim. f tX∩K′ 0 implies Df(x) is of full rank for all

x ∈ X ∩K ′ ∩ f−1(0), where Df = (∂fi/∂xj)ij. Since X ∩K ′ is compact, one can

9

find K ′′ compact such that (X ∩K ′) ⊂ (K ′′ − ∂K ′′) and Df is of full rank on K ′′.

Let U := X ∩K ′ ∩ g−1(0). By taking |y| sufficiently small, one has U ⊂ K ′′. Then

for each x ∈ U , (∂gi∂xj

)ij

=

(∂fi∂xj

− yi ·∂λi

∂xj

)ij

is of full rank by choosing |y| small enough again (view det(Df) as a continuous

function). Hence the claim.

Remark. In the above proof, the verification of g tK 0 can also be carried

out as we do in proving g tK′∩X 0. In this case, it is much easier because of

K ∩ g−1(0) = f−1(y). By our assumption, det(Df) = 0 on f−1(y). In fact, these

two proofs are the same but write in different ways.

Definition. (Thom Spaces)

Let ξ be a vector bundle over a smooth manifold M . Define the Thom space of ξ

to be T(ξ) := D(ξ)/S(ξ), where D(ξ) contains all elements in ξ with length ≤ 1,

and S(ξ) contains all elements in ξ with length = 1. Let t0 be the point of T(ξ)

identified by S(ξ).

Fact. Suppose a smooth manifold M is also a CW-complex. If ξ is a k-dimensional

vector bundle over M , then T(ξ) is a (k − 1)-connected CW-complex.

Theorem 2. Let ξ be a vector bundle of a closed manifold M of rank k. Then there

is a group homomorphism τ : πn+k(T(ξ)) → Ωn.

Proof. There are several steps.

For convenience, denote T(ξ) by T. Given a map f : Sn+k → T, one can ap-

proximate f by a map f0 : Sn+k → T on f−10 (T−t0) = f−1(T−t0). Choose an open

covering W1, · · · ,Wr of compact set f−10 (M) such that f0(Wi) is contained in some

π−1(Ui) ∼= Ui × Rk, where Ui are some open subsets of M . Let ρi : π−1(Ui) → Rk

be the projection. Pick compact Ki ⊂ Wi such that f−10 (M) ⊂ (K1 ∪ · · · ∪ Kr).

Our strategy is to modify f0 within one Wi after another, and then define the

last one to be our desired function. We want to construct maps f1, · · · , fr satisfying:

(a) Every fi is smooth in f−1i (T−t0) and fi |Wi−Ki

= fi−1 |Wi−Ki.

(b) fi tK1∪···∪Kr M for i = 1, · · · , r.

10

(c) π(fi(x)) ∈ M equals π(f0(x)) for all x ∈ f−10 (T−t0).

We start from f0 and construct fi inductively. Assume fi−1 has already been con-

structed. Condition (c) implies that fi−1(Wi) ⊂ π−1(Ui) ∼= Ui ×Rk. Also, it follows

from condition (b) that the map ρi fi−1 : Wi → Rk has

ρi fi−1 t(K1∪···∪Ki)∩Wi0.

By lemma 1, we can approximate ρi fi−1 : Wi → Rk by a map ρi fi : such that

(a′) ρi fi |Wi−Ki= ρi fi−1 |Wi−Ki

.

(b′) fi tK1∪···∪Kr 0 N := g−1(T(ξ)− t0).

So we can define fi : Wi → π−1(Ui) ∼= Ui × Rk whose first coordinate π(fi(x)) is

determined by condition (c) and the second coordinate ρi fi(x) is determined by

condition (a), (a′) and (b′). It is then clear that condition (b) holds. Hence we

define f1, f2, · · · , fr inductively.

Now let g := fr. We must prove the following claim.

Claim: g−1(M) ⊂ (K1 ∪ · · · ∪Kr).

Indeed, g tK1∪···∪Kr M . If we can prove the claim, then we will conclude g t M .

Now we prove the claim. Since K1∪· · ·Kr is a compact neighborhood of f−10 (M)

in the compact manifold Sn+k, one can find c ∈ (0, 1) such that |f0(x)| < c for all

x /∈ (K1 ∪ · · · ∪Kr). Let fi is chosen to satisfy

|fi(x)− fi−1(x)| <c

rfor all x ∈ Sn+k .

Consequently, |g(x) − f0(x)| < c for all x ∈ Sn+k and thus |g(x)| = 0 for any

x /∈ (K1 ∪ · · · ∪ Kr). That is, g−1(M) ⊂ (K1 ∪ · · · ∪ Kr). Hence g t M . So

we naturally define τ([f ]) = [g−1(M)], where g−1(M) is a compact manifold of

dimension n by our construction.

Next we have to check that this map is well defined, i.e., [f0] = [f1] ∈ πn+k(T(ξ))

such that f0 t M and f1 t M implies f−10 (M) = f−1

1 (M). We take a smooth map

F : Sn+k ×[0, 1] → T(ξ) such that

F (x, [0,1

3]) = f0(x) F (x, [

2

3, 1]) = f1(x).

Since f0 t M and f1 t M , we have

F tSn+k ×(0, 13]∪N×[ 2

3,1) M.

11

By lemma 1 and similar process of above construction, we can approximate F by

F ′ : Sn+k ×[0, 1] → T(ξ) such that

F ′ tSn+k ×(0,1) and F ′(x, [0, δ)) = f0(x), F ′(x, (1− δ, 1]) = f1(x),

where δ is some positive number less than 1/3. Then ∂F ′(M) = f−11 (M)− f−1

0 (M).

Hence [f−10 (M)] = [f−1

1 (M)] ∈ Ωn.

The final thing is to verify τ is definitely a group homomorphism. It is obvious

that the addition in homotopy group corresponds to the disjoint union in cobordism

group. Hence we construct the homomorphism.

Recall in section 1 we have defined the oriented canonical vector bundle

γkp := γk(Rk+p) over Gk(Rk+p).

Lemma 2. If k ≥ n and p ≥ n, then the homomorphism τ : π(T(γkp )) → Ωn is

surjective.

Proof. Let Mn be a compact smooth manifold of dimension n. By Whitney em-

bedding theorem, one can embed Mn into Rn+k for some k. Let TNk be the normal

vector bundle of Mn in Rn+k (the superscript indicates that TN is a k-dimensional

vector space). By the existence of tubular neighborhood of Mn, there exists a

neighborhood U of Mn in Rn+k diffeomorphic to TNk. Thus

U ∼= TNk Gauss map−−−−−−−→ γkn → γk

p

canonical map−−−−−−−−−→ T(γkp ).

Let g : U → T(γkp ) be the resulting map. There is no doubt that g t M and

g−1(Gk(Rk+p)) = M . Extend g to a map g : Sn+k → T(γkp ) by viewing Sn+k ∼=

Rn+k ∪∞ and sending Sn+k −U to t0. Hence

[g] ∈ π(T(γkp )) and [Mn] = [g−1(Gk(Rk+p))]

Corollary 1. The manifolds

CP2i1 × · · · × CP2ir ,

where (i1, · · · , ir) ∈ P(m), are independent, i.e., free of relations, in Ω4m. Hence

Ω4m has rank ≥ p(m).

12

Theorem 3. Let X be a finite CW complex which is r-connected with r ≥ 1. Then

we have the isomorphism πn(X)⊗Q ∼= Hn(X;Q) for n ≤ 2r.

Theorem 4. (Thom)

Ω⊗Q ∼= Q[CP2,CP4,CP6, · · · ].

Proof. By lemma 2, we know that Ωn is a homomorphic image of πn+k(T(γkp )). By

theorem 3, we have

πn+k(T(γkp ))⊗Q ∼= Hn+k(T(γk

p );Q).

Note that we have the natural isomorphism Hn+k(T(γkp );Q) ∼= Hn+k(T(γk

p );Q). By

theorem 1, we have

rankΩn ≤ p(m) if n = 4m

rankΩn = 0 if 4 - n.

However, corollary 1 tells us that rankΩn ≥ p(m) when n = 4m. As a result,

rankΩn = p(m) when n = 4m. In fact, corollary 1 implies more:

CP2i1 × · · · × CP2ir (i1, · · · , ir) ∈ P(m)

is a set of basis of Ω4m ⊗Q. Hence the Thom’s theorem.

13

3 Hirzebruch’s Signature Formula

We first recall some properties about homology and cohomology of manifolds.

Fact. Let F be a field. If M is a n-dimensional compact oriented manifold, then

Hn(M ;F ) ∼= F

Hm(M ;F ) = 0 for all m > n.

Theorem 5. (Poincaré Duality)

If M is a n-dimensional oriented compact manifold, then

Hk(M ;R) = Hn−k(M ;R)

for k = 0, 1, · · · , n, where R can be any coefficient ring.

We consider the pairing

Hi(M ;F )×Hn−i(M ;F ) → F.

Since dimH i(M ;F ) = dimHn−i(M ;F ) = dimHn−i(M ;F ) by Poincaré duality and

natural isomorphism of vector spaces, we can view Hn−i(M ;F ) as the dual space of

H i(M ;F ). In particular, we have

Hi(M ;Z)×Hn−i(M ;Z) → Z .

For n being even, we can define a non-degenerate bilinear form on Hn/2(M ;Z) by

⟨x, y⟩ := x(y),

where x, y ∈ Hn/2(M ;Z) and x ∈ Hn/2(M ;Z) is the isomorphic image of x. Note

that ⟨ ·, · ⟩ is symmetric if n/2 is even an is alternating if n/2 is odd.

Definition. (Fundamental Classes)

Let M be a n-dimensional compact oriented manifold. The fundamental (homology)

class of M , denoted by µM , is the generator of H(M ;Z) ∼= Z, which is compatible

with the orientation of M .

14

Now given a compact oriented manifold M of dimension 4n. For any two

α, β ∈ H2n(M,R), we define the pairing

⟨α, β⟩ := (α β)(µM).

This pairing is non-degenerate. Recall that deRham theorem tells us that the deR-

ham cohomology is isomorphic singular cohomology, i.e.,

HpdR(M ;R) ∼= Hp(M ;R).

In some situations, the viewpoint of deRham cohomology is easier to compute than

singular cohomology.

Definition. (Signatures of Compact Oriented Smooth Manifolds)

Let M be a compact oriented smooth manifold of dimension n. If 4 - n, then its

signature, denoted by σ(M), is defined to be zero. If n = 4k, then σ(M) is defined

to be the signature of the symmetric bilinear form ⟨ ·, · ⟩ on H2k(M ;R).

Remark.

(a) Recall that the signature of a symmetric real bilinear form is the difference of

the number of positive eigenvalues and the number of negative eigenvalues.

(b) We now have two bilinear forms: one on homology, the other on cohomology.

However, we use the same notation ⟨ ·, · ⟩ because the signature of a compact oriented

smooth manifold can be defined to be the signature of ⟨ ·, · ⟩ on homology or ⟨ ·, · ⟩

on cohomology.

For short, σ : Ω → Z is a map between rings. As we expected, σ is actually a

ring homomorphism. We state and prove this result in the following theorem, which

was first presented in Thom’s paper.

Theorem 6. (Thom)

σ : Ω → Z is a ring homomorphism, i.e., σ satisfies

(a) σ(M +N) = σ(M) + σ(N).

(b) σ(M ×N) = σ(M)× σ(N).

(c) σ(M) = 0 if M = ∂W .

15

Proof. Let dimM = m, dimN = n and dimW = m+ 1.

(a) This is obvious.

(b) Let V := M × N . If 4 - dimV , then 4 - m or 4 - n. Thus σ(M × N) = 0 and

σ(M)× σ(N) = 0.

Now suppose dimV = 4k. By Künneth theorem,

H2k(V ;R) ∼=⊕

s+t=2k

Hs(M ;R)⊗R H t(N ;R).

Two elements x, y ∈ H2k(V ;R) are said to be orthogonal if xy(µV ) := x y(µV ) =

0. Let vsi , wtj be basis of Hs(M ;R), H t(N ;R) such that vsi v

m−sj = δij, w

tiw

n−tj =

δij for s = m/2, t = n/2. Let A = Hm/2(M ;R)⊕Hn/2(N ;R) if m,n are both even

and A = 0 for other cases. Define B := A⊥ in H2k(V ;R). So

vsi ⊗ wtj | s+ t = 2k, s = m

2, t = n

2

is an orthogonal basis of B.

(c) The coefficient ring of the following diagram is R.

· · · H2k+1(W4k+1,M4k) H2k(M

4k) H2k(W4k+1) · · ·

· · · H2k(W4k+1) H2k(M4k) H2k+1(W 4k+1,M4k) · · ·

∂∗

≀

i∗

≀ ≀

i∗ δ∗

Note that imi∗ is of half dimension of H2k(M). For any two cocycles x, y in M4k

are obtained by restricting cocycles x′, y′ in W 4k+1, i.e., i∗(x′) = x, i∗(y′) = y. Then

⟨x, y ⟩ = (x y)(µM) = i∗(x′ y′)(µM) = (x′ y′)i∗(µM).

Thus the number of positive and negative eigenvalues are the same.

Let A =⊕

n∈N0An be a graded ring. Define AΠ := a0+a1+a2+ · · · | ai ∈ Ai.

Particularly, we are interested in AΠ1 := 1 + a1 + a2 + · · · | ai ∈ Ai ⊂ AΠ.

Definition. (Multiplicative Sequences)

Let x ∈ AΠ. We say that Knn∈N0 is a multiplicative sequence if

(i) each Kn(x1, · · · , xn) is a homogeneous polynomial of degree n.

(ii) K(ab) = K(a)K(b) for any a, b ∈ AΠ1 , where

K(x) := 1 +K1(x1) +K2(x1, x2) + · · · .

16

Proposition 2. Let A be the graded polynomial ring R[t] where t is a variable of

degree 1. Given f(t) ∈ 1+ λ1t+ λ2t2 + · · · ∈ R[[t]], there is an unique multiplicative

sequence Kn such that K(1 + t) = f(t).

Proof. By definition, we want to find Kn satisfying

K(1 + t) = 1 +K1(t) +K2(t, 0) +K3(t, 0, 0) + · · · = 1 + λ1t+ λ2t2 + λ3t

3 · · · .

That is, we want to find Kn(x1, · · · , xn) whose coefficient of xn1 -term is λn for each

n.

We prove the existence of Kn at first. Fix n ∈ N. Let t1, · · · , tn be

algebraically independent and all of degree 1. For I = (i1, · · · , ir) ∈ P(n), de-

fine λI := λi1 · · ·λir . Let s1, · · · , sn be the elementary symmetric polynomials of

t1, · · · , tn. Note that s1, · · · , sn is also algebraically independent. We claim

that

Kn(s1, · · · , sn) :=∑

I∈P(n)

λIgI(s1, · · · , sn)

is the desired multiplicative sequence. The definition of gI is as following:

gI(s1, · · · , sn) =∑

ti1j1 · · · tirjr

with 1 ≤ j1, · · · , jr ≤ n all distinct and no ”repeated terms”. 1 It is clear that we

have the formula

gI(ab) =∑HJ=I

gH(a)gJ(b).

Hence K(ab) = K(a)K(b).

Remark. In the case of proposition 2, we call the multiplicative sequence Kn

belongs to the formal power series f(t).

Now we define the action of multiplicative sequence Kn on m-dimensional

compact oriented smooth manifold Mm. If 4 - m, define K(Mm) = 0. If m = 4k,

define

K(M4k) := Kk(p1, · · · , pk)(µM).

1See chapter 16 of [MS] for the complete definition of gI . Notice that the notations of [MS] are

different from this article.

17

Theorem 7. (Hirzebruch’s Signature Formula)

Let Ln be the multiplicative sequence belonging to√t

tanh√t= 1 +

1

3t− 1

45t2 + · · ·+ (−1)n−122nBn

(2n)!tn + · · · ,

where Bk denotes the k-th Bernoulli number. Then σ(M4k) = L(M4k).

Proof. By Thom’s cobordism theorem, we only need to check that σ(CP2k) =

L(CP2k) for each k ∈ N. We have already computed that σ(CP2k) = 1. To

compute L(CP2k), we recall that p(CP2k) = (1 + a2)2k+1, where a := −c1(γ1) with

γ1 the canonical line bundle of CP2k. By definition,

L(1 + a2 + 0 + · · · ) =√a2

tanh√a2

=⇒ L(p(CP2k)) =( a

tanh a

)2k+1

.

Now we replace a by a complex variable z. We want to compute the coefficient of

z2k in the Taylor expansion of( z

tanh z

)2k+1

. The substitution u = tanh z with

dz =du

1− u2

gives1

2πi

∫dz

(tanh z)2k+1=

1

2πi

∫1 + u2 + u4 + · · ·

u2k+1du = 1.

Hence L(CP2 k) = 1.

18

4 Construction of Exotic 7-Spheres

We recall some definitions and properties in Morse theory.

Definition. (Morse Function)

A Morse function f on a manifold M is a real-valued function whose critical points

(the points where the first derivative of f vanishes) are all non-degenerate, i.e., its

Hessian matrix is non-singular.

Note that the index of a non-degenerate critical point y of f is the dimension of

the largest subspace of the tangent space of M at y on which the Hessian is negative

definite.

Lemma 3. (Morse Lemma)

Let y be a non-degenerate critical point of f : Mn → R with index α. Then there

exists a chart (x1, x2, · · · , xn) in a neighborhood U of y such that

xi(y) = 0 ∀1 ≤ i ≤ n

and

f(x) = f(b)− x21 − · · · − x2

α + x2α+1 + · · ·+ x2

n ∀x ∈ U \ y.

Recall that we have introduced the SO(4)-vector bundles over S4 of rank 4. Let

ξhj be the vector bundle of rank 4 defined by fhj : S3 → SO(4) where fhj(v)w :=

vhwvj (By viewing v ∈ S3 → H, the multiplication vhwvj is doing in H). Let Shj be

the sphere bundle of ξhj. Now we are ready to construct the exotic seven spheres.

Idea: Suppose h + j = 1.We will show that M7k := Shj is a topological 7-sphere

by constructing a Morse function on M7k , where k is an odd number and assume

h − j = k. Finally, if we can show that M7k is not diffeomorphic to standard S7,

then we complete the construction of exotic 7-spheres by explicitly constructing a

exotic 7-sphere, M7k . The final step will be carried out by computing characteristic

classes.

Step 1. If f : M7k → R is a Morse function with two critical points. Let y0, y1 be

the two critical points. Since M7k is compact, the two critical points are actually the

19

maximun and minimun of f . By rescaling the function f , we may assume f(y0) = 0

and f(y1) = 1. Now consider the gradient flow:

dxdt

= ∇f(x).

Note that this flow is orthogonal to each level set f−1(a) for a ∈ (0, 1). Hence we

have f−1([0, a]) ∼= f−1([0, b]) for each a, b ∈ (0, 1). For a sufficiently small, it follows

from Morse lemma that there exists a chart (x1, · · · , x7) of neighborhood f−1([0, a])

of y0 such that

f(x) = x21 + · · ·+ x2

7.

That is, f−1([0, a]) ∼= D7. Therefore, f−1([0, 1)) = M7k −y1 ∼= D7. Hence 7k ∼= S7

topologically.

Step 2. Now our mission is to construct a Morse function and apply step 1 to

conclude that 7k ∼= S7 topologically. To construct a Morse function on M7k , we

have to realize M7k as a more understandable structure. We will check in this step

that M7k can be realized as gluing two copies of R4×S3 along (R4−0)× S3 via a

diffeomorphism g of (R4−0)× S3 given by

g : (u, v) → (u′, v′)

(u

|u|2,uhvuj

|u|

).

(Notice that the operation are done in H for R4−0 → H and S3 → H.) This map

is well-defined because of h + j = 1. Now take the case u = u′ into consideration.

In this case, we get a restricted map of g on S3:

g := S3 → SO(4)

g(u)v := uhvuj.

This is exactly the map that we define M7k . Hence the result.

Step 3. From step 2, we have two coordinate charts (u, v) and (u′′, v′). In this step,

we will verify that

f(u, v) =Re(v)√1 + |u|2

=Re(u′′)√1 + |u′′|2

where u′′ := u′(v′)−1 =u

|u| · uhvuj

is our desired Morse function on M7k , i.e., it has two non-degenerate critical points.

First of all, direct computation shows that the second equality holds. It is clear that

20

our definition of f forces f to be an increasing function in the first variable under

the chart (u′′, v′). As a result, the critical points lie in the chart (u, v), and thus the

critical points look like (0, v). However, f reduce to be height function of S3 in this

case. Hence the critical points are (0, 1) and (0,−1).

Step 4. We want to find some condition of k that will make M7k not diffeomorphic

to standard S7. Now suppose that M7k is diffeomorphic to standard S7. We can

attach a standard 8-dimensional disk to D(ξhj) along M7k because we assume that

M7k∼= S7. Denote the resulting 8-dimensional space by W 8

k . By our construction,

W 8k∼= T(ξhj). As a consequence,

H i(S4) ∼= H4+i(D(ξhj),Shj) ∼= H4+i(T(ξhj), t0).

This implies

H i(W 8k )

∼= Z if i = 0, 4, 8

H i(W 8k ) = 0 if i = 0, 4, 8.

So the signature of W 8k equals 1 or −1 up to our choice of the orientation of W 8

k .

Assume σ(W 8k ) = 1. Now Hirzebruch’s signature formula gives 1 =

7p2 − p2145

. Our

task now turns to compute the Pontryajin classes of W 8k . Recall that we have

computed in section 1 that e(ξhj) = u and p1(ξhj) = 2k u, where u := e(γ1R) ∈

H4(HP1,Z) (γ1R is the canonical line bundle of HP1). Let π : ξhj → S4 be the

canonical projection. We have Tξhj ∼= π∗(T S4)⊕ π∗(ξhj), and then apply Whitney

sum formula and naturality of characteristic classes to obtain (note that p(T S4) = 1)

p(Tξhj) = π∗p(ξhj)

p1(Tξhj) = π∗p1(ξhj) = π∗(2k u) = 2k u = 2ke(ξhj).

Hence p21(TW8k ) = p21(Tξhj) = 4k2. Finally, write 4k2 + 45 = 7p2 ≡ 0 ( mod 7)

and use this to conclude k ≡ 0 ( mod 7). However, our assumption is that k is

any odd number. That is, the equality k ≡ 0 ( mod 7) fails to hold for any odd k.

Hence we conclude that our original hypothesis is wrong: M7k is not diffeomorphic

to standard S7.

21

Bibliography

[DFN1] B. A. Dubrovin, A. T. Fomenkov and S. P. Novikov. Modern Geometry -

Methods and Applications Part I. The Geometry of Surfaces, Transforma-

tion Groups, and Fields. Springer-Verlag, Graduate Texts in Mathematics

93, 1984.

[DFN2] B. A. Dubrovin, A. T. Fomenkov and S. P. Novikov. Modern Geometry

- Methods and Applications Part III. Introduction to Homology Theory.

Springer-Verlag, Graduate Texts in Mathematics 124, 1984.

[DFN3] B. A. Dubrovin, A. T. Fomenkov and S. P. Novikov. Modern Geometry

- Methods and Applications Part III. Introduction to Homology Theory.

Springer-Verlag, Graduate Texts in Mathematics 124, 1984.

[Hi] F. Hirzebruch. Topological Methods in Algebraic Geometry. Springer-

Verlag, Classics in Mathematics, 1995.

[Mi] J. W. Milnor. Topology from the Differentiable Viewpoint. Princeton Uni-

versity Press, 1965.

[MS] J. W. Milnor and J. D. Stasheff. Characteristic Classes. Princeton Univer-

sity Press, 1974.

[Wa] C.-L. Wang. A Report On Hirzebruch Signature Formula and Milnor’s

Exotic Seven Spheres. Electronic source, available at http://www.math.

ntu.edu.tw/~dragon/Lecture%20Notes/har-minor.pdf.

Note.

The outline of this article is mainly based on [Wa], where some contents are

referred to other books.

22